Abstract
In this paper, we introduce the notion of exponentially p-convex function and exponentially s-convex function in the second sense. We establish several Hermite–Hadamard type inequalities for exponentially p-convex functions and exponentially s-convex functions in second sense. The present investigation is an extension of several well known results.
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1 Introduction
Recently, the study of convex functions has become more important due to variety of their nature. Many generalizations of this notion have been established. For more details see [1,2,3,4,5,6, 13, 16,17,18,19].
Convex functions satisfy many integral inequalities. Among these, the Hermite–Hadamard inequality is well known. The Hermite–Hadamard inequality [14, 15] for a convex function \(\psi : \mathcal{K}\rightarrow \mathbb{R}\) on an interval \(\mathcal{K}\) is
for all \(u_{1},u_{2}\in \mathcal{K}\) with \(u_{1}< u_{2}\). Many authors have made generalizations to inequality (1.1). For more results and details, see [3, 4, 6, 7, 17,18,19, 22, 24,25,26].
Definition 1.1
Consider an interval \(\mathcal{K}\subset (0, \infty )= \mathbb{R}_{+}\), and \(p\in \mathbb{R\setminus }\{0\}\). A function \(\psi :\mathcal{K}\rightarrow \mathbb{R}\) is called p-convex, if
for all \(u_{1},u_{2}\in \mathcal{K}\) and \(r\in [0,1]\). If the inequality in (1.2) is reversed, then ψ is called p-concave.
Example 1.1
A function \(\psi :(0,\infty )\rightarrow \mathbb{R}\), defined by \(\psi (u)=u^{p}\) for \(p\in \mathbb{R\setminus }\{0\}\), is p-convex as well as p-concave.
Iscan [19] gave the following results.
Theorem 1.2
([19])
Consider an interval \(\mathcal{K}\subset (0,\infty )\), and \(p\in \mathbb{R\setminus }\{0\}\). Let \(\psi :\mathcal{K}\rightarrow \mathbb{R}\) be p-convex and \(u_{1},u_{2}\in \mathcal{K}\), \(u_{1}< u_{2}\). If \(\psi \in L_{1}[u_{1},u_{2}]\), then we have
Lemma 1.1
([19])
Let \(\psi :\mathcal{K}\rightarrow \mathbb{R}\) be a differentiable function on \(\mathcal{K}^{\circ }\), i.e., the interior of \(\mathcal{K}\), and \(u_{1},u_{2}\in \mathcal{K}\), \(u_{1}< u_{2}\), and \(p\in \mathbb{R\setminus }\{0\}\). If \(\psi '\in L_{1}[u_{1},u_{2}]\), then
Definition 1.2
([16])
Let \(s\in (0,1]\). A function \(\psi :\mathcal{K}\subset \mathbb{R}_{0}\rightarrow \mathbb{R}_{0}\), where \(\mathbb{R}_{0}=[0, \infty )\), is called s-convex in the second sense, if
for all \(u_{1},u_{2}\in \mathcal{K}\) and \(r\in [0,1]\).
Example 1.3
A function \(\psi :(0,\infty )\rightarrow (0,\infty )\), defined by \(\psi (u)=u^{s}\) for \(s\in (0,1)\), is s-convex in the second sense.
Dragomir et al. [8, 9] gave the following important results.
Theorem 1.4
([9])
Let \(s\in (0,1)\) and \(\psi :\mathbb{R}_{0}\rightarrow \mathbb{R}_{0}\) be s-convex in the second sense. Let \(u_{1},u_{2} \in [0,\infty )\), \(u_{1}\leq u_{2}\). If \(\psi \in L_{1}[u_{1},u_{2}]\), then
Lemma 1.2
([8])
Let \(\psi :\mathcal{K} \rightarrow \mathbb{R}\) be a differentiable mapping on \(\mathcal{K}^{\circ }\), the interior of \(\mathcal{K}\), and \(u_{1},u_{2}\in \mathcal{K}\) be two distinct points. If \(\psi '\in L_{1}[u_{1},u_{2}]\), then
Awan et al. [4] introduced the following new class of convex functions.
Definition 1.3
([4])
A function \(\psi : \mathcal{K}\subseteq \mathbb{R}\rightarrow \mathbb{R}\) is called exponentially convex, if
for all \(u_{1},u_{2}\in \mathcal{K}\), \(r\in [0,1]\) and \(\alpha \in \mathbb{R}\). If the inequality in (1.8) is reversed, then ψ is called exponentially concave.
Example 1.5
A function \(\psi :\mathbb{R}\rightarrow \mathbb{R}\), defined by \(\psi (u)=-u^{2}\), is an exponentially convex for all \(\alpha >0\).
The Beta and Hypergeometric functions are defined as:
and
respectively, see [21].
2 Exponentially p-convex functions
Now we introduce exponentially p-convex functions.
Definition 2.1
Consider an interval \(\mathcal{K}\subset (0, \infty )=\mathbb{R}_{+}\) and \(p\in \mathbb{R\setminus }\{0\}\). A function \(\psi :\mathcal{K} \rightarrow \mathbb{R}\) is called exponentially p-convex, if
for all \(u_{1},u_{2}\in \mathcal{K}\), \(r\in [0,1]\) and \(\alpha \in \mathbb{R}\). If the inequality in (2.1) is reversed, then ψ is called exponentially p-concave.
It is easy to note that, by taking \(\alpha =0\), an exponentially p-convex function becomes p-convex.
Example 2.1
Consider a function \(\psi :(\sqrt{2},\infty )\rightarrow \mathbb{R}\), defined by \(\psi (u)=(\ln (u))^{p}\) for \(p\geq 2\). Then ψ is exponentially p-convex for all \(\alpha <0\), and not p-convex.
Note that ψ satisfies inequality (2.1) for all \(\alpha <0\). But for \(u_{1}=2\), \(u_{2}=3\) and \(p=5\), inequality (1.2) does not hold.
2.1 Integral inequalities
Throughout this section, we denote by \(\mathcal{K}\subset (0, \infty )=\mathbb{R}_{+}\) an interval with interior \(\mathcal{K}^{\circ }\) and \(p\in \mathbb{R\setminus }\{0\}\). We start with our results for exponentially p-convex functions.
Theorem 2.2
Let \(\psi :\mathcal{K}\rightarrow \mathbb{R}\) be an integrable exponentially p-convex function. Let \(u_{1},u_{2}\in \mathcal{K}\) with \(u_{1}< u_{2}\). Then for \(\alpha \in \mathbb{R}\), we have
where
Proof
By using the exponential p-convexity of ψ, we have
Letting \(w^{p}=ru_{1}^{p}+(1-r)u_{2}^{p}\) and \(z^{p}=(1-r)u_{1}^{p}+ru _{2}^{p}\), we get
Integrating with respect to \(r\in [0,1]\) and applying a change of variable, we find
Hence the first inequality of (2.2) has been established. For the next inequality, again using the exponential p-convexity of ψ, we have
Integrating with respect to \(r\in [0,1]\), we get
By combining (2.5) and (2.7), we get (2.2). □
Remark 2.1
In Theorem 2.2, by taking \(\alpha =0\), we attain inequality (1.3) in Theorem 1.2.
Theorem 2.3
Let \(\psi :\mathcal{K}\rightarrow \mathbb{R}\) be a differentiable function on \(\mathcal{K}^{\circ }\) and \(u_{1},u_{2} \in \mathcal{K}\) with \(u_{1}< u_{2}\) and \(\psi '\in L_{1}[u_{1},u_{2}]\). If \(|\psi '|^{q}\) is exponentially p-convex on \([u_{1},u_{2}]\) for \(q\geq 1\) and \(\alpha \in \mathbb{R}\), then
where
Proof
Applying the power mean inequality to (1.4) of Lemma 1.1, we get
Since \(|\psi '|^{q}\) is exponentially p-convex on \([u_{1},u_{2}]\), we have
It is easy to note that
Hence the proof is completed. □
Remark 2.2
In Theorem 2.3,
-
(a)
by taking \(\alpha =0\), we attain Theorem 7 in [19];
-
(b)
by taking \(p=1\), we attain Theorem 5 in [4].
Corollary 2.4
Let \(\psi :\mathcal{K}\rightarrow \mathbb{R}\) be a differentiable function on \(\mathcal{K}^{\circ }\) and \(u_{1},u_{2} \in \mathcal{K}\), \(u_{1}< u_{2}\), and \(\psi '\in L_{1}[u_{1},u_{2}]\). If \(|\psi '|\) is exponentially p-convex on \([u_{1},u_{2}]\), then
where \(B_{2}\) and \(B_{3}\) are given in Theorem 2.3.
Remark 2.3
In Corollary 2.4,
-
(a)
by taking \(\alpha =0\), we attain Corollary 1 in [19];
-
(b)
by taking \(p=1\), we attain Theorem 3 in [4].
Theorem 2.5
Let \(\psi :\mathcal{K}\rightarrow \mathbb{R}\) be a differentiable function on \(\mathcal{K}^{\circ }\). Let \(u_{1},u_{2} \in \mathcal{K}\), \(u_{1}< u_{2}\), and \(\psi '\in L_{1}[u_{1},u_{2}]\). If \(|\psi '|^{q}\) is exponentially p-convex on \([u_{1},u_{2}]\), and \(q,l>1\), \(1/q+1/l=1\), and \(\alpha \in \mathbb{R}\), then
where
Proof
Using Hölder’s inequality on (1.4) of Lemma 1.1 and then applying the exponential p-convexity of \(|\psi '|^{q}\) on \([u_{1},u_{2}]\), we get
where after calculations, we have
□
Remark 2.4
In Theorem 2.5,
-
(a)
by letting \(\alpha =0\), we attain Theorem 8 in [19];
-
(b)
by letting \(p=1\), we attain Theorem 4 in [4].
Theorem 2.6
Let \(\psi :\mathcal{K}\rightarrow \mathbb{R}\) be a differentiable function on \(\mathcal{K}^{\circ }\) and \(u_{1},u_{2} \in \mathcal{K}\), \(u_{1}< u_{2}\), and \(\psi '\in L_{1}[u_{1},u_{2}]\). If \(|\psi '|^{q}\) is exponentially p-convex on \([u_{1},u_{2}]\), and \(q,l>1\), \(1/q+1/l=1\), and \(\alpha \in \mathbb{R}\), then
where
Proof
Using Hölder’s inequality on (1.4) of Lemma 1.1 and then applying the exponential p-convexity of \(|\psi '|^{q}\) on \([u_{1},u_{2}]\), we get
where a simple calculation implies
and
By substituting (2.16) and (2.17) into (2.15), we get (2.14). □
Remark 2.5
In Theorem 2.6, by letting \(\alpha =0\), we attain Theorem 9 in [19].
2.2 Applications
Consider some special means of two positive numbers \(u_{1}\), \(u_{2}\), \(u_{1}< u_{2}\):
-
(1)
The arithmetic mean
$$ A=A(u_{1},u_{2})=\frac{u_{1}+u_{2}}{2}; $$ -
(2)
The harmonic mean
$$ H=H(u_{1},u_{2})=\frac{2u_{1}u_{2}}{u_{1}+u_{2}}; $$ -
(3)
The p-logarithmic mean
$$ L_{p}=L_{p}(u_{1},u_{2})= \biggl( \frac{u_{2}^{p+1}-u_{1}^{p+1}}{(p+1)(u _{2}-u_{1})} \biggr)^{\frac{1}{p}} ,\quad p\in \mathbb{R}\setminus \{-1,0 \}. $$
In the next three propositions we consider \(0< u_{1}< u_{2}\) and \(q>1\).
Proposition 2.1
Let \(\alpha \in \mathbb{R}\) and \(p<1\). Then we have
where \(B_{6}\) is defined as in Theorem 2.6.
Proof
The proof ensues from Theorem 2.6, for a function \(\psi :(0, \infty )\rightarrow \mathbb{R}\), \(\psi (w)=\frac{1}{w}\). Here note that \(|\psi '(w)|^{q}=|\frac{1}{w^{2}}|^{q}\) is exponentially p-convex for all \(p<1\) and \(\alpha \in \mathbb{R}\). □
Proposition 2.2
Let \(\alpha \leq 0\) and \(p>1\). Then we have
where \(B_{6}\) is defined as in Theorem 2.6.
Proof
The proof ensues from Theorem 2.6, for \(\psi :(0,\infty )\rightarrow \mathbb{R}\), \(\psi (w)=w^{p}\). Here note that \(|\psi '(w)|^{q}=|pw ^{p-1}|^{q}\) is exponentially p-convex for all \(p>1\) and \(\alpha \leq 0\). □
Proposition 2.3
Let \(\alpha \leq 0\) and \(p>1\). Then we have
where \(B_{6}\) is given as in Theorem 2.6.
Proof
The proof ensues from Theorem 2.6, for \(\psi :(0,\infty )\rightarrow \mathbb{R}\), \(\psi (w)=w\). Here note that \(|\psi '(w)|^{q}=1\) is exponentially p-convex for all \(p>1\) and \(\alpha \leq 0\). □
3 Exponentially s-convex functions in the second sense
We first generalize Definition 1.2.
Definition 3.1
Let \(s\in (0,1]\) and \(\mathcal{K}\subset \mathbb{R}_{0}\) be an interval. A function \(\psi :\mathcal{K}\rightarrow \mathbb{R}\) is called exponentially s-convex in the second sense, if
for all \(u_{1},u_{2}\in \mathcal{K}\), \(r\in [0,1]\) and \(\alpha \in \mathbb{R}\). If the inequality in (3.1) is reversed then ψ is called exponentially s-concave.
Observe that, by taking \(\alpha =0\), an exponentially s-convex function becomes s-convex.
Example 3.1
Consider a function \(\psi :[0,\infty )\rightarrow \mathbb{R}\), defined by \(\psi (u)=\ln (u)\) for \(s\in (0,1)\). Then ψ is exponentially s-convex, for all \(\alpha \leq -1\), but not s-convex in the second sense.
3.1 Integral inequalities
Throughout this section, we denote by \(\mathcal{K}\subset \mathbb{R} _{0}\) an interval with nonempty interior \(\mathcal{K}^{\circ }\) and \(s\in (0,1]\). We start our new results with the following theorem.
Theorem 3.2
Let \(\psi :\mathcal{K}\subset \mathbb{R}_{0}\rightarrow \mathbb{R}\) be an integrable exponentially s-convex function in the second sense on \(\mathcal{K}^{\circ }\). Then for \(u_{1},u_{2}\in \mathcal{K}\) with \(u_{1}< u_{2}\) and \(\alpha \in \mathbb{R}\), we have
where
Proof
Applying exponential s-convexity of ψ, we have
Letting \(w=ru_{1}+(1-r)u_{2}\) and \(z=(1-r)u_{1}+ru_{2}\), we get
Integrating with respect to \(r\in [0,1]\) and applying a change of variable, we find
Hence the proof of the first inequality of (3.2) has been completed. For the next inequality, again using the exponential s-convexity of ψ, we have
Integrating with respect to \(r\in [0,1]\), we get
By combining (3.5) and (3.7), we get (3.2). □
Remark 3.1
In Theorem 3.2, by letting \(\alpha =0\), we get inequality (1.6) in Theorem 1.4.
Theorem 3.3
Let \(\psi :\mathcal{K}\rightarrow \mathbb{R}\) be a differentiable function on \(\mathcal{K}^{\circ }\) and \(u_{1},u_{2} \in \mathcal{K}\) with \(u_{1}< u_{2}\) and \(\psi '\in L_{1}[u_{1},u_{2}]\). If \(|\psi '|\) is exponentially s-convex in the second sense on \([u_{1},u_{2}]\), then
Proof
From Lemma 1.2, we have
Using the exponential s-convexity of \(\psi '\), we get
Since
by substituting equalities (3.11) and (3.12) into (3.10), we get inequality (3.8). □
Corollary 3.4
Under the assumptions of Theorem 3.3, we have the following:
-
(a)
If \(s=1\), then
$$\begin{aligned} &\biggl\vert \frac{\psi (u_{1})+\psi (u_{2})}{2}-\frac{1}{u_{2}-u_{1}} \int _{u_{1}}^{u_{2}}\psi (w)\,dw \biggr\vert \\ &\quad \leq \frac{u_{2}-u_{1}}{12} \biggl[7 \biggl\vert \frac{\psi '(u_{1})}{e^{\alpha u_{1}}} \biggr\vert +5 \biggl\vert \frac{\psi '(u_{2})}{e^{\alpha u_{2}}} \biggr\vert \biggr] . \end{aligned}$$(3.13) -
(b)
If \(\alpha =0\), then
$$\begin{aligned} & \biggl\vert \frac{\psi (u_{1})+\psi (u_{2})}{2}-\frac{1}{u_{2}-u_{1}} \int _{u_{1}}^{u_{2}}\psi (w)\,dw \biggr\vert \\ &\quad \leq \frac{u_{2}-u_{1}}{2(s+1)(s+2)} \bigl[(3s+4) \bigl\vert \psi '(u _{1}) \bigr\vert +(s+4) \bigl\vert \psi '(u_{2}) \bigr\vert \bigr] . \end{aligned}$$(3.14)
Theorem 3.5
Let \(\psi :\mathcal{K}\rightarrow \mathbb{R}\) be a differentiable function on \(\mathcal{K}^{\circ }\) and \(u_{1},u_{2} \in \mathcal{K}\), \(u_{1}< u_{2}\), and \(\psi '\in L_{1}[u_{1},u_{2}]\). If \(|\psi '|\) is exponentially s-convex in the second sense on \([u_{1},u_{2}]\), then
Proof
From Lemma 1.2, we have
Using the exponential s-convexity of \(\psi '\), we get
It is easily seen that
Thus by substituting equalities (3.18) and (3.19) into (3.17), we achieve inequality (3.15). □
Remark 3.2
In Theorem 3.5,
-
(a)
by taking \(\alpha =0\), we obtain Theorem 1, for \(q=1\), in [23];
-
(b)
by taking \(s=1\), we obtain Theorem 3 in [4].
Theorem 3.6
Let \(\psi :\mathcal{K}\rightarrow \mathbb{R}\) be a differentiable function on \(\mathcal{K}^{\circ }\) and \(u_{1},u_{2} \in \mathcal{K}\) with \(u_{1}< u_{2}\) and \(\psi '\in L_{1}[u_{1},u_{2}]\). If \(|\psi '|^{q}\) is exponentially s-convex in the second sense on \([u_{1},u_{2}]\) with \(q> 1\), then we have
Proof
From Lemma 1.2, we have
Applying the power-mean inequality, we find
Since \(|\psi '|^{q}\) is exponentially s-convex, we get
where
Using (3.22)–(3.24) in (3.21), we get (3.20). □
Remark 3.3
In Theorem 3.6,
-
(a)
by putting \(\alpha =0\), we get Theorem 1, for \(q>1\), in [23];
-
(b)
by putting \(s=1\), we get Theorem 5 in [4].
Theorem 3.7
Let \(\psi :\mathcal{K}\rightarrow \mathbb{R}\) be a differentiable function on \(\mathcal{K}^{\circ }\) and \(u_{1},u_{2} \in \mathcal{K}\) with \(u_{1}< u_{2}\) and \(\psi '\in L_{1}[u_{1},u_{2}]\). If \(|\psi '|^{q}\) is exponentially s-convex in the second sense on \([u_{1},u_{2}]\) and \(q,l> 1\), \(\frac{1}{l}+\frac{1}{q}=1\), then we have
Proof
From Lemma 1.2 and using Hölder’s inequality, we have
Since \(|\psi '|^{q}\) is exponentially s-convex, we get
Hence the proof is completed. □
Remark 3.4
In Theorem 3.7,
-
(a)
by letting \(\alpha =0\), we get
$$\begin{aligned} & \biggl\vert \frac{\psi (u_{1})+\psi (u_{2})}{2}-\frac{1}{u_{2}-u_{1}} \int _{u_{1}}^{u_{2}}\psi (w)\,dw \biggr\vert \\ &\quad \leq \frac{u_{2}-u_{1}}{2(l+1)^{ \frac{1}{l}}} \biggl[ \frac{ \vert \psi '(u_{1}) \vert ^{q} + \vert \psi '(u _{2}) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}}; \end{aligned}$$(3.28) -
(b)
by letting \(s=1\), we get Theorem 4 in [4].
3.2 Applications
Suppose d is a partition of the interval \([u_{1},u_{2}]\), that is, \(d: u_{1}=w_{0}< w_{1}<\cdots <w_{m-1}<w_{m}=u_{2}\), then the trapezoidal formula is given as
We known that if \(\psi :[u_{1},u_{2}]\rightarrow \mathbb{R}\) is twice differentiable on \((u_{1},u_{2})\) and \(\mathcal{M}= \max_{w\in (u_{1},u_{2})}|\psi ''(w)|<\infty \), then
where the remainder term is given as
It is noticed that if \(\psi ''\) does not exist or \(\psi ''\) is unbounded, then (3.29) is invalid. However, Dragomir and Wang [10,11,12] have shown that the term \(R(\psi ,d)\) can be obtained by using the first derivative only. These estimates surely have several applications. In this section, we estimate the remainder term \(R(\psi ,d)\) in a new sense.
Proposition 3.1
Let \(\psi :\mathcal{K}\subseteq \mathbb{R}_{0}\rightarrow \mathbb{R}\) be a differentiable function on \(\mathcal{K}^{\circ }\). Let \(u_{1},u_{2}\in \mathcal{K}\), \(u_{1}< u_{2}\). If \(|\psi '|\) is exponentially s-convex in the second sense on \([u_{1},u_{2}]\) and \(s\in (0,1]\), then in (3.29), for every partition d of \([u_{1},u_{2}]\), we have
Proof
Applying Theorem 3.5 on the subinterval \([w_{n},w_{n+1}]\) (\(n=0,1,\ldots ,m-1\)) of the partition d, we obtain
Summing over n from 0 to \(m-1\), we get
□
Proposition 3.2
Let \(\psi :\mathcal{K}\subseteq \mathbb{R}_{0}\rightarrow \mathbb{R}\) be a differentiable function on \(\mathcal{K}^{\circ }\) and \(u_{1},u_{2}\in \mathcal{K}\) with \(u_{1}< u_{2}\). If \(|\psi '|^{q}\) is exponentially s-convex in the second sense on \([u_{1},u_{2}]\) and \(s\in (0,1]\) and \(q,l> 1\) such that \(\frac{1}{l}+\frac{1}{q}=1\), then in (3.29), for every partition d of \([u_{1},u_{2}]\), we have
Proof
Using Theorem 3.7 and similar arguments as in Proposition 3.1, we get the required result. □
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The present investigation is supported by National University of Science and Technology (NUST), Islamabad, Pakistan.
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Mehreen, N., Anwar, M. Hermite–Hadamard type inequalities for exponentially p-convex functions and exponentially s-convex functions in the second sense with applications. J Inequal Appl 2019, 92 (2019). https://doi.org/10.1186/s13660-019-2047-1
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DOI: https://doi.org/10.1186/s13660-019-2047-1